I presume you're talking about Irish cream liqueur such as Bailey's. That doesn't have any coffee in it! - basically it's cream, sugar and Irish whiskey, and it's about half whiskey so you would only have a small glass. Bailey's is 17% alcohol, so rather stronger than wine. The manufacturing process involves homogenising it, like milk, or the cream would separate out from the whiskey - "oil and water don't mix!" Same reason making mayonnaise needs a lot of hard whisking as you add the oil VERRRRRY slowly - it's the only way to keep the oil and vinegar blended, and you put egg yolk in it to help that happen.
Talking of which, I remember when milk wasn't homogenised so the cream floated on the top and you had to shake the bottle to mix it in. Unless you bought red top homogenised milk, which didn't taste as nice so Mum and Dad bought silver top. (Grandma and Grandad bought gold top Channel Islands milk, which has more cream in it! Jersey and Guernsey cows just naturally produce fattier milk.) These days it's all homogenised, and dairies are obviously doing it a better way than how red top was made. It's all changed so much in just 40 years - we had milk delivered every day by the milkman on his electric milk float, the milkman called on Saturday to get paid, and when I got older and Mum and Dad left me "home alone" on a Saturday morning, they left behind the money so I could pay the bill.
It was even better for the environment - leave out your "empties" and he'd collect the glass bottles to take back to the dairy to wash out and use again.
Personally I find Bailey's a bit too sickly sweet and I'd rather have an Irish coffee, which is black coffee, sugar, a shot of Irish whiskey and topped with cream poured carefully over the back of a spoon so it floats on top. You get a longer drink that way, too. You drink the hot coffee through the cold cream. Now that's obviously a sensation you like as you like affogato.
BTW affogato means "drowned" and I know this from Nigella Lawson, who I would totally trust on his because she studied Italian at Oxford. Drown the ice cream in coffee - it makes sense to call it that, doesn't it?
What I have also seen in restaurants is other alcoholic coffees. Irish is the original but you could have Gaelic coffee with Scotch whisky in it, Jamaican coffee with rum, any "hard liquor" would work.